Perspective Projections

Centered on truncated octahedron

The following image shows a perspective projection of the runcitruncated
24-cell into 3D, centered on a truncated octahedron:

The nearest truncated octahedron (red) to the 4D viewpoint lies in the
center of this projection. Eight other truncated octahedra (yellow) lie
adjacent to it, in cubic formation. The green cells are rhombicuboctahedra.
For clarity, cells that lie on the far side are omitted, along with the
triangular prisms and hexagonal prisms.

Centered on rhombicuboctahedron

The next image shows another projection of the runcitruncated 24-cell, this
time centered on a rhombicuboctahedron:

The nearest cell to the 4D viewpoint here is the blue rhombicuboctahedron at
the center of the projection image. It is surrounded by 6 truncated octahedra
(yellow). Another 8 rhombicuboctahedra (green) straddle these truncated
octahedra. Again, for clarity's sake, the triangular and hexagonal prisms are
omitted, as are the cells that lie on the far side of the runcitruncated
24-cell.

Notice that this layout of cells corresponds with the layout in the previous
projection, with the roles of the rhombicuboctahedra and the truncated
octahedra swapped. This is a consequence of the fact that the
rhombicuboctahedral cells correspond with the cells of a 24-cell, and the
truncated octahedra correspond with the cells of a dual 24-cell. Since
the 24-cell is self-dual, these two types of cells are laid out in the same
way, and exhibit equivalent relationships with each other.

Parallel Projections

The following image shows a projection of the runcitruncated 24-cell,
centered on a truncated octahedron. This time, we use a parallel projection for
exploring the structure of the polytope.

The truncated octahedron in the center is the closest cell to the 4D
viewpoint. For clarity, we render all other cells in a light transparent
color.

As we have seen in the perspective projections, six rhombicubotahedra
surround this nearest cell, as shown next:

These rhombicuboctahedra look a bit flattened, because they are being viewed
from a 45° angle. In 4D, they are perfectly uniform. They each share a square
face with the nearest cell.

The hexagonal faces of the nearest cell are shared with 8 hexagonal prisms,
shown next:

Touching the edges of the nearest cell are 12 triangular prisms:

Here are the preceding cells, all together:

The exposed hexagonal faces of the 8 hexagonal prisms are joined to another
8 truncated octahedra, shown next:

The gaps between these cells are obviously filled by hexagonal prisms:

The exposed triangular faces of the rhombicuboctahedra joined to another 24
triangular prisms:

Straddled between these triangular prisms are also another 24 hexagonal
prisms:

All the cells seen so far lie on the near side of the runcitruncated
24-cell. The next layer of cells lie on the “equator”: they are
oriented 90° from the 4D viewpoint. Among these equatorial cells are 6
truncated octahedra, shown below:

For clarity, we omit the other cells that we have seen before. These
truncated octahedra have been foreshortened into octagons, because they are
being viewed from a 90° angle. In 4D, they are perfectly uniform truncated
octahedra.

These truncated octahedra are bridged by 12 rhombicuboctahedra, shown
next:

Again, these rhombicuboctahedra are perfectly uniform in 4D, but they have
been foreshortened into octagons because they are at a 90° angle with the
line-of-sight.

These rhombicuboctahedra themselves are bridged by 8 hexagonal prisms, as
seen below:

These hexagonal prisms have been foreshortened into hexagons.

The remaining square gaps in the runcitruncated 24-cell's equator are filled
in by triangular prisms, 24 of them:

These triangular prisms, of course, have also been foreshortened; they
appear as rectangles. They each share an edge with a red truncated octahedron.
The apparent edges they share with the green rhombicuboctahedra are actually
the foreshortened images of their triangular faces. The apparent edges they
share with the hexagonal prisms are actually the foreshortened images of one of
their square faces.

Past this point are the cells on the far side of the runcitruncated 24-cell.
They follow an identical layout with the cells on the near side that we have
already seen, except in reverse order of layers.

Therefore, in summary, on the near side we have 1+8=9 truncated octahedra, 6
rhombicuboctahedra, 8+12+24=44 hexagonal prisms, and 12+24=36 triangular
prisms. On the equator we have 6 truncated octahedra, 12 rhombicuboctahedra, 8
hexagonal prisms, and 24 triangular prisms. On the far side, we have again 9
truncated octahedra, 6 rhombicuboctahedra, 44 hexagonal prisms, and 36
triangular prisms. Thus, in total, there are 9+6+9=24 truncated octahedra,
6+12+6=24 rhombicuboctahedra, 44+8+44=96 hexagonal prisms, and 36+24+36=96
triangular prisms, corresponding with the cells, ridges, edges, and vertices of
a 24-cell, from which the runcitruncated 24-cell is
derived.

Coordinates

The coordinates for the runcitruncated 24-cell are all permutations of
coordinates and changes of sign of:

This site is best viewed with
ANY browser. You do not need
a specific version of a specific browser to view this site; all
standards-compliant browsers are fully capable of accessing all the content on
this site. No unpopular browsers were harmed in the creation of this site.